find-f-a-dx-1-ax-2-with-a-from-R-

Question Number 36818 by maxmathsup by imad last updated on 06/Jun/18

f i n d f (a) = \int \frac{d x}{\sqrt{1 - {a x}^{2}}} w i t h a f r o m R .

Commented by prof Abdo imad last updated on 06/Jun/18

c a s e 1 a > 0 c h a n g e m e n t \sqrt{a} x = s i n t g i v e

f (a) = \int \frac{1}{c o s t} \frac{c o s t d t}{\sqrt{a}} = \frac{1}{\sqrt{a}} a r c s i n (\sqrt{a} x) + c

c a s e 2 a < 0 c h a n g e m e n t \sqrt{- a} x = s h (t) g i v e

f (a) = \int \frac{1}{c h (t)} \frac{c h (t)}{\sqrt{- a}} = \frac{1}{\sqrt{- a}} a r g s h (\sqrt{- a} x)

= \frac{1}{\sqrt{- a}} l n {x \sqrt{- a} + \sqrt{1 - {a x}^{2}}) + c .

Answered by MJS last updated on 06/Jun/18

\int \frac{d x}{\sqrt{1 - {a x}^{2}}} =

[t = \sqrt{a} x \to d x = \frac{d t}{\sqrt{a}}]

= \frac{1}{\sqrt{a}} \int \frac{d t}{\sqrt{1 - t^{2}}} = \frac{1}{\sqrt{a}} \arcsin t =

\frac{\sqrt{a}}{a} \arcsin \sqrt{a} x + C

{\begin{cases} a > 0 \frac{1}{\sqrt{a}} \arcsin \sqrt{a} x + C \\ a = 0 x + C \\ a < 0 \frac{1}{\sqrt{- a}} arcsinh \sqrt{- a} x + C \end{cases}

Facebook Tweet Pin